An arXiv preprint gives necessary-and-sufficient tests for when QK—a specialized class of analytic functions on the unit disc—can be embedded into L2(μ) for a positive Borel measure μ. It treats both bounded and compact embeddings, then applies the criteria to Volterra integral operators and multiplication operators.
The work is theoretical: it ranges over mathematical measures, analytic functions and operators rather than an empirical sample. In plain language, it asks when a measure gives enough control for the QK-to-L2(μ) map to stay bounded, and what extra condition is needed for compactness.
The capacity test
The analysis fixes a polar b-admissible dyadic resolution of the disc, essentially a radial-and-angular grid for organizing the proof. It constructs two capacity tests: a discrete dyadic capacity and a semidefinite-programming capacity, built through a matrix-based optimization framework. The paper proves that the two are comparable for the finite positive measures in its stated setting, using conic duality and the complex Grothendieck inequality.
What happens near the boundary
For the zero-at-origin subspace, the trace embedding into L2(μ) is bounded exactly when the discrete dyadic capacity is finite. For the full QK space, boundedness requires finite total measure mass as well as finite semidefinite-programming capacity.
Compactness is controlled by the edge of the disc. On the zero-at-origin subspace, it requires finite capacity and a boundary-shell restriction whose capacity tends to zero as ρ approaches 1 from below. For the full QK space, finite total mass is required too, along with finite capacity and the same vanishing boundary-shell condition.
The same framework reaches operators
The framework transfers to the Volterra operator T_g. The paper says T_g is bounded exactly when g belongs to QK and the capacity of the associated measure μ_{g,a,K} stays uniformly finite as a ranges over the unit disc.
For compactness, boundedness is not enough: the associated boundary-shell capacities must tend to zero uniformly over a as ρ approaches 1 from below.
Multiplication operators receive a parallel test. M_g is bounded on QK exactly when g belongs to both H∞ and QK—the paper’s bounded-analytic class and the QK space—and satisfies the uniform capacity condition.
A mathematical result with a defined scope
The authors interpret these theorems as closing the gap between earlier sufficient and necessary conditions for T_g and as providing a complete solution to the multiplier problem. That is an interpretation stated by the authors, rather than an independent comparison supplied in the review.
These are mathematical results within a defined setting: K∈WQ, analytic functions on the unit disc, positive Borel measures and the chosen polar dyadic resolution. The supplied analysis does not address whether the criteria extend beyond K∈WQ or beyond the unit disc.
The document is version 1 of an arXiv preprint dated 20 August 2026. Because it uses no empirical observations or dataset, there are no study participants, effect sizes or sampling uncertainty to report.
Paper data and sources
Original title: Carleson measures, Volterra integral operators and multipliers for $Q_K$ spaces
Authors: Wujun Cao, Zhouyuan Jiang, Songxiao Li
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text